User blog:Edwin Shade/Pop The Balloons Challenge III (Continuation of Googleaarex's Challenge)
In this post, Googology user Googleaarex created a challenge in which he asked us to calculate the value of PRBGY according to the rules that he laid out. This is a continuation of that challenge, but with more balloons, the rules are as follows: 1.) A "wave" of balloons may be either linear or planar; the starting score is set at 0. 2.) a.) When a red balloon pops it increases the score by 1. b.) When a blue balloon pops it spawn S red balloons, where S is the current score. These balloons are spawned diagonally. c.) When a green balloon pops, it spawns S blue balloons diagonally. d.) When a yellow balloon pops, it spawns S green balloons diagonally. e.)When a rainbow balloon pops, it spawns S S-level color balloons diagonally. 3.) When a white balloon pops, all the balloons in the wave to the right of the white balloon are copied S times and placed to the left of the white balloon, which then dissappears; it's vacancy is filled depending on the configuration of the array or linear array. 4.) Red balloons constitute "1st-level" color ballons, blue balloons constitute "2nd-level" color balloons, and so forth arbitrarily high. 5.) Red balloons are denoted by an R, blue balloons B, green balloons G, yellow balloons Y, and rainbow balloons C. White balloons are denoted by a W, and the pin is notated by a P. 6.) If a balloon is popped and leaves a vacancy, all balloons above that balloon are to be dropped down by one in that row. If a balloon creates a vacancy but there are no balloons above it, then all balloons in the wave to the right of that balloon are to be move one space to the left. Here is an example of how we would deal with the following array of balloons. CR PRBGG S=0 First, the red balloon ® can be popped giving us a score of 1. All the other balloons are now moved one space to the left as the space once occupied by the red balloon is now vacant. CR PBGG S=1 Next we can replace the blue balloon (B) with S red balloons, or just one red balloon, This adds 1 again to our score and leaves us with the following array CR PGG S=2 Now is where things become interesting, because we are left with a green balloon (G) in front of the pin, meaning we now pop the green balloon and insert 2 blue balloons in it's place. Now is where I will show what I meant when I mentioned diagnol spawning. R CB --> CB PBG PBGR S=2 The red balloon that was in the top right corner was pushed out of the way by the two blue balloons, and thus falls down to ground level, (ground level is on level with the pin), extending the array by an extra row. The next array will show this as well. B CR --> CRB PRGR PRGR S=2 Next we pop the red balloon leaving a vertical gap in the array, which is filled in by the rainbow balloon ©, which drops down from above. RB PCGR S=3 Now the rainbow balloon is popped, spawning 3 3rd-level color balloons in a diaganol. This corresponds to 3 green balloons. G GB PGGRR S=3 Let's follow this array a bit more. B BBG PBGRRGG S=3 RB RBGBB PRGRRGGGGG S=3 RB RBGBB PGRRGGGGG S=4 Clearly, the array is growing larger, but must terminate no matter how large it may be when solving. Also, because balloons are spawned diaganolly rather to the direct right, and are subject to gravity, it means that one cannot get exact values for a given balloon array using the fast-growing hierchy; because the next array depends explicitly on the configuration of the last array. With this in mind, I challenge you to find the tighest minimum and maximum bounds you can for the score values of the following arrays.Note that if you have found the exact value of an array, that is even better. Level I: Classic PRBGY Level II: A Classical Twist PRBGYCW Level III: Bull's Eye ! RRRRRRRRR RBBBBBBBR RBGGGGGBR RBGYYYGBR RBGYCYGBR RBGYYYGBR RBGGGGGBR RBBBBBBBR PRRRRRRRRR Have fun ! Category:Blog posts